Change the chapter
Question
When an electron and positron annihilate, both their masses are destroyed, creating two equal energy photons to preserve momentum. (a) Confirm that the annihilation equation $\textrm{e}^+ + \textrm{e}^- \to \gamma + \gamma$ conserves charge, electron family number, and total number of nucleons. To do this, identify the values of each before and after the annihilation. (b) Find the energy of each $\gamma$ ray, assuming the electron and positron are initially nearly at rest. (c) Explain why the two $\gamma$ rays travel in exactly opposite directions if the center of mass of the electron-positron system is initially at rest.
Question by OpenStax is licensed under CC BY 4.0.
Final Answer
Please see the solution video.
Solution Video

OpenStax College Physics Solution, Chapter 31, Problem 29 (Problems & Exercises) (3:14)

Sign up to view this solution video!

View sample solution
Video Transcript

This is College Physics Answers with Shaun Dychko. When a positron and an electron annihilate together, they produce two <i>γ</i>-rays, or two photons. And we are going to check to make sure charge, electron family number and number of nucleons is conserved in this reaction. So considering charge, we have a charge of plus 1 on the positron and charge of negative 1 on the electron which is a total of zero and on the right-hand side, the <i>γ</i>-rays each have a charge of zero and so that checks out. The electron family number is positive 1 for a positron and negative 1 for an electron which is a total of zero and the <i>γ</i>-rays also have an electron family number of zero and so this checks out. And there are no nucleons on either side of this equation and so that is conserved zero equals zero. In part (b), we are asked to figure out the energy of each of the photons. Now we know that the total momentum of these two photons has to be zero because the positron and electron we assume are initially or at least there's the center of mass of the system of positron and electron is not moving and so there's zero momentum to begin with which means there has to be zero momentum after the annihilation. And so that means momentum of the first <i>γ</i>-ray photon has to equal the negative of the momentum of the second <i>γ</i>-ray photon and which is to say that they're their momenta are of equal magnitude and so we'll call that just <i>p</i> with no subscript since their values are same just in opposite direction, that's what the negative sign there means. Now we have this equation here telling us the energy of a photon in terms of its momentum; that is equation 30 from chapter 29 and the energy of the <i>γ</i>-ray photon then is its momentum times <i>c</i> which is the same as the energy of the other photon since they each have the same momentum. OK. So the total energy of the two photons together is gonna be the first one plus the second one but since they are both the same, we'll just say 2 times the energy of each photon and so that's 2 times the mass energy of an electron or positron but those are the same anyway. And so divide both sides by 2 and the reason we have this 2 here by the way is because there is the mass energy of the positron plus the mass energy of the electron but both of those are the same and so we just say 2 times the mass energy of one of them. And so we have the energy then is the mass of the electron times <i>c</i> squared which is 0.511 megaelectron volts and that is the energy of each photon. And part (c) says why do they travel in exactly opposite directions and the answer is to conserve momentum so that the total momentum when you add the momenta as vectors equals zero.